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# Minimum operations to transform given string to another by moving characters to front or end

Given two Strings S and T of length N consisting of lowercase alphabets, which are permutations of each other, the task is to print the minimum number of operations to convert S to T. In one operation, pick any character of the string S and move it either to the start or end of the string S.

Examples:

Input: S = “abcde”, T = “edacb”
Output: 3
Explanation:
We can convert S to T in 3 moves:
1. move ‘d’ to start: “dabce”
2. move ‘e’ to start: “edabc”
3. move ‘b’ to end: “edacb”

Input: S = “dcdb”, T = “ddbc”
Output: 1
Explanation:
Move ‘c’ to end

Naive Approach: The naive approach is to try all possibilities of swapping a character. One can put some character to the front, to the end, or can leave it in the same position. The above three operations can be solved using recursion and print the minimum number of steps required after all the steps.
Time Complexity: O(3N), where N is the length of the given string.
Auxiliary Space: O(1)

Efficient Approach: To optimize the above approach, the idea is to observe that after moving the characters of the string S, the unchanged characters come together to form a contiguous substring in T. So, if we can maximize the length of this subsequence, then the count of operations to convert string S to T is:

N – length of the longest contiguous substring of T that is a subsequence of S

Therefore, to find the length of the longest contiguous substring of T that is a subsequence of string S, find the longest common subsequence of S and T. Let dp[][] stores the length of the longest contiguous substring of T that is a subsequence of string S, . Now dp[i][j] will store the length of the longest suffix of T[0, …, j] that is also a subsequence of S[0, …, i]. The recurrence relation is given by:

• If i is greater than 0, dp[i][j] = max(dp[i-1][j], dp[i][j]).
• If S[i] is equals to T[i] then, dp[i][j] = 1 + dp[i-1][j-1].

Below is the implementation of the above approach:

## C++

 // C++ program for the above approach#include using namespace std; int dp[1010][1010]; // Function that finds the minimum number// of steps to find the minimum characters// must be moved to convert string s to tint solve(string s, string t){     int n = s.size();     // r = maximum value over all    // dp[i][j] computed so far    int r = 0;     // dp[i][j] stores the longest    // contiguous suffix of T[0..j]    // that is subsequence of S[0..i]    for (int i = 0; i < n; i++) {         for (int j = 0; j < n; j++) {             dp[i][j] = 0;            if (i > 0) {                 dp[i][j] = max(dp[i - 1][j],                               dp[i][j]);            }            if (s[i] == t[j]) {                 int ans = 1;                if (i > 0 && j > 0) {                     ans = 1 + dp[i - 1][j - 1];                }                 // Update the maximum length                dp[i][j] = max(dp[i][j], ans);                r = max(r, dp[i][j]);            }        }    }     // Return the resulting length    return (n - r);} // Driver Codeint main(){    // Given string s, t    string s = "abcde";    string t = "edacb";     // Function Call    cout << solve(s, t);    return 0;}

## Java

 // Java program for the above approachclass GFG{    static int[][] dp = new int[1010][1010];     // Function that finds the minimum number    // of steps to find the minimum characters    // must be moved to convert String s to t    static int solve(String s, String t)    {        int n = s.length();         // r = maximum value over all        // dp[i][j] computed so far        int r = 0;         // dp[i][j] stores the longest        // contiguous suffix of T[0..j]        // that is subsequence of S[0..i]        for (int i = 0; i < n; i++)        {            for (int j = 0; j < n; j++)            {                dp[i][j] = 0;                if (i > 0)                {                    dp[i][j] = Math.max(dp[i - 1][j],                                        dp[i][j]);                }                if (s.charAt(i) == t.charAt(j))                {                    int ans = 1;                    if (i > 0 && j > 0)                    {                        ans = 1 + dp[i - 1][j - 1];                    }                     // Update the maximum length                    dp[i][j] = Math.max(dp[i][j], ans);                    r = Math.max(r, dp[i][j]);                }            }        }         // Return the resulting length        return (n - r);    }     // Driver Code    public static void main(String[] args)    {        // Given String s, t        String s = "abcde";        String t = "edacb";         // Function Call        System.out.print(solve(s, t));    }} // This code is contributed by shikhasingrajput

## Python3

 # Python3 program for the above approachdp = [[0] * 1010] * 1010 # Function that finds the minimum number# of steps to find the minimum characters# must be moved to convert string s to tdef solve(s, t):     n = len(s)     # r = maximum value over all    # dp[i][j] computed so far    r = 0     # dp[i][j] stores the longest    # contiguous suffix of T[0..j]    # that is subsequence of S[0..i]    for j in range(0, n):        for i in range(0, n):            dp[i][j] = 0                         if (i > 0):                dp[i][j] = max(dp[i - 1][j],                               dp[i][j])                         if (s[i] == t[j]):                ans = 1                if (i > 0 and j > 0):                    ans = 1 + dp[i - 1][j - 1]                                 # Update the maximum length                dp[i][j] = max(dp[i][j], ans)                r = max(r, dp[i][j])                     # Return the resulting length    return (n - r) # Driver Code # Given string s, ts = "abcde"t = "edacb" # Function callprint(solve(s, t)) # This code is contributed by code_hunt

## C#

 // C# program for the above approachusing System;class GFG{static int[, ] dp = new int[1010, 1010]; // Function that finds the minimum number// of steps to find the minimum characters// must be moved to convert String s to tstatic int solve(String s, String t){    int n = s.Length;     // r = maximum value over all    // dp[i, j] computed so far    int r = 0;     // dp[i, j] stores the longest    // contiguous suffix of T[0..j]    // that is subsequence of S[0..i]    for (int i = 0; i < n; i++)    {        for (int j = 0; j < n; j++)        {            dp[i, j] = 0;            if (i > 0)            {                dp[i, j] = Math.Max(dp[i - 1, j],                                    dp[i, j]);            }            if (s[i] == t[j])            {                int ans = 1;                if (i > 0 && j > 0)                {                    ans = 1 + dp[i - 1, j - 1];                }                 // Update the maximum length                dp[i, j] = Math.Max(dp[i, j], ans);                r = Math.Max(r, dp[i, j]);            }        }    }     // Return the resulting length    return (n - r);} // Driver Codepublic static void Main(String[] args){            // Given String s, t    String s = "abcde";    String t = "edacb";     // Function Call    Console.Write(solve(s, t));}} // This code is contributed by shikhasingrajput

## Javascript



Output

3

Time Complexity: O(N2), where N is the length of the given string
Auxiliary Space: O(N2)

Efficient approach :  Space optimization using 2 vectors

In this approach we use two vectors because in previous approach we can see that dp[i][j] is only dependent on the current row and previous row of dp.

dp[i][j] = max(dp[i – 1][j], dp[i][j]);

Implementation Steps :

• Made 2 vectors says curr and prev that use to keep track of values of current and previous row of matrix respectively.
• Now change dp[i] to curr and dp[i-1] to prev in previous approach.
• After every iteration of outer loop store all values of curr to prev and move to the next iterations

Implementation :

## C++

 // C++ program for the above approach#include using namespace std;   // Function that finds the minimum number// of steps to find the minimum characters// must be moved to convert string s to tint solve(string s, string t){     int n = s.size();         // vector made to track only 2 rows of 2d dp    vectorprev(n+1);    vectorcurr(n+1);         // r = maximum value over all    // dp[i][j] computed so far    int r = 0;     // dp[i][j] stores the longest    // contiguous suffix of T[0..j]    // that is subsequence of S[0..i]    for (int i = 0; i < n; i++) {         for (int j = 0; j < n; j++) {             curr[j] = 0;            if (i > 0) {                 curr[j] = max(prev[j], curr[j]);            }            if (s[i] == t[j]) {                 int ans = 1;                if (i > 0 && j > 0) {                     ans = 1 + prev[j - 1];                }                                 // Update the maximum length                curr[j] = max(curr[j], ans);                r = max(r, curr[j]);            }        }        // store current row in previous row and do another iterations        prev = curr;    }         // Return the resulting length    return (n - r);} // Driver Codeint main(){    // Given string s, t    string s = "abcde";    string t = "edacb";     // Function Call    cout << solve(s, t);    return 0;} // this code is contributed by bhardwajji

## Java

 // Java program for the above approach import java.util.*; public class Main {         // Function that finds the minimum number    // of steps to find the minimum characters    // must be moved to convert string s to t    static int solve(String s, String t) {        int n = s.length();        int[] prev = new int[n+1];        int[] curr = new int[n+1];        int r = 0;         // dp[i][j] stores the longest contiguous suffix of T[0..j]        // that is subsequence of S[0..i]        for (int i = 0; i < n; i++) {            for (int j = 0; j < n; j++) {                curr[j] = 0;                if (i > 0) {                    curr[j] = Math.max(prev[j], curr[j]);                }                if (s.charAt(i) == t.charAt(j)) {                    int ans = 1;                    if (i > 0 && j > 0) {                        ans = 1 + prev[j - 1];                    }                    curr[j] = Math.max(curr[j], ans);                    r = Math.max(r, curr[j]);                }            }            // store current row in previous row and do another iterations            prev = curr.clone();        }                 // Return the resulting length        return (n - r);    }         // Driver Code    public static void main(String[] args) {        // Given string s, t        String s = "abcde";        String t = "edacb";                 // Function Call        System.out.println(solve(s, t));    }} // Contributed by sdeadityasharma

## Python3

 # Function that finds the minimum number# of steps to find the minimum characters# must be moved to convert string s to tdef solve(s, t):    n = len(s)     # list made to track only 2 rows of 2d dp    prev = [0] * (n + 1)    curr = [0] * (n + 1)     # r = maximum value over all    # dp[i][j] computed so far    r = 0     # dp[i][j] stores the longest    # contiguous suffix of T[0..j]    # that is subsequence of S[0..i]    for i in range(n):        for j in range(n):            curr[j] = 0            if i > 0:                curr[j] = max(prev[j], curr[j])            if s[i] == t[j]:                ans = 1                if i > 0 and j > 0:                    ans = 1 + prev[j - 1]                                 # Update the maximum length                curr[j] = max(curr[j], ans)                r = max(r, curr[j])                 # store current row in previous row and do another iterations        prev = curr[:]         # Return the resulting length    return (n - r) # Driver Codeif __name__ == '__main__':    # Given string s, t    s = "abcde"    t = "edacb"     # Function Call    print(solve(s, t))

## C#

 using System; public class MainClass {    // Function that finds the minimum number    // of steps to find the minimum characters    // must be moved to convert string s to t    static int Solve(string s, string t)    {        int n = s.Length;        int[] prev = new int[n + 1];        int[] curr = new int[n + 1];        int r = 0;         // dp[i][j] stores the longest contiguous suffix of        // T[0..j] that is subsequence of S[0..i]        for (int i = 0; i < n; i++) {            for (int j = 0; j < n; j++) {                curr[j] = 0;                if (i > 0) {                    curr[j] = Math.Max(prev[j], curr[j]);                }                if (s[i] == t[j]) {                    int ans = 1;                    if (i > 0 && j > 0) {                        ans = 1 + prev[j - 1];                    }                    curr[j] = Math.Max(curr[j], ans);                    r = Math.Max(r, curr[j]);                }            }            // store current row in previous row and do            // another iterations            prev = (int[])curr.Clone();        }         // Return the resulting length        return (n - r);    }     // Driver Code    public static void Main(string[] args)    {        // Given string s, t        string s = "abcde";        string t = "edacb";         // Function Call        Console.WriteLine(Solve(s, t));    }}// This code is contributed by user_dtewbxkn77n

## Javascript

 // Javascript program for the above approach // Function that finds the minimum number// of steps to find the minimum characters// must be moved to convert string s to tfunction solve(s, t) {  const n = s.length;  let prev = new Array(n + 1).fill(0);  let curr = new Array(n + 1).fill(0);  let r = 0;   // dp[i][j] stores the longest contiguous suffix of T[0..j]  // that is subsequence of S[0..i]  for (let i = 0; i < n; i++) {    for (let j = 0; j < n; j++) {      curr[j] = 0;      if (i > 0) {        curr[j] = Math.max(prev[j], curr[j]);      }      if (s.charAt(i) == t.charAt(j)) {        let ans = 1;        if (i > 0 && j > 0) {          ans = 1 + prev[j - 1];        }        curr[j] = Math.max(curr[j], ans);        r = Math.max(r, curr[j]);      }    }    // store current row in previous row and do another iterations    prev = curr.slice();  }   // Return the resulting length  return n - r;} // Driver Codeconst s = "abcde";const t = "edacb"; // Function Callconsole.log(solve(s, t));

Output

3

Time Complexity: O(N2), where N is the length of the given string
Auxiliary Space: O(N) only use 1d vector not 2d matrix to store values.

Note: The above naive approach is efficient for smaller strings whereas, the above efficient approach is efficient for larger strings.